Answer
Yes.
Work Step by Step
Consider $1 - \dfrac{x^2 y^2}{3} \lt \dfrac{\tan^{-1} (xy)}{xy} \lt 1$
Since, $\lim\limits_{(x,y) \to (0,0) } (1 - \dfrac{x^2 y^2}{3})=1$
and $\lim\limits_{(x,y) \to (0,0) } (1)=1$
Yes, the limit for $\lim\limits_{(x,y) \to (0,0) } (\dfrac{\tan^{-1} (xy)}{xy} )=1$ by the Sandwich Theorem.
